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Use geometric conditions to compute lengths, areas and angles without trusting a sketch.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Choose and apply geometric relationships from stated information while distinguishing linear and area scale factors.
A straight angle measures 180 degrees and a full turn 360 degrees. Perpendicular lines meet at right angles; parallel lines do not meet in the plane. A triangle's interior angles sum to 180 degrees. The Pythagorean theorem applies specifically to a right triangle: the sum of the squares of its legs equals the square of its hypotenuse. Review multiplication, square roots and solving equations before combining these relationships in a multistep geometry question.
| Term | What it means |
|---|---|
| Congruent | Equal in shape and size, so corresponding lengths and angles agree. |
| Similar | Equal in shape, with corresponding angles equal and corresponding lengths in one ratio. |
| Altitude | A perpendicular distance from a vertex to the line containing a chosen base. |
| Sector | The region of a circle between two radii and their intercepted arc. |
Geometry problems provide relationships through words, labels and stated properties. A triangle that looks isosceles is not necessarily isosceles unless equal sides or angles are given or can be deduced. A corner that appears square does not establish a right angle. Use the sketch to organize supplied information, then base each calculation on a justified relationship. Coordinate axes and data graphs have different conventions from an unscaled geometric sketch; read the displayed scale when the problem supplies one.
Before selecting a formula, name the requested quantity and its dimension. A length uses linear units, an area square units and a volume cubic units. A perimeter can remain fixed while area changes, so one measurement does not automatically determine another. If a quantitative comparison leaves a shape free to vary, consider two valid configurations instead of measuring the picture.
Another way: steps
List stated properties; identify the target dimension; choose justified relationships; compute; check feasibility and units.
Vertical angles formed by two intersecting lines are equal, and adjacent angles on a straight line are supplementary. When a transversal crosses parallel lines, corresponding angles and alternate interior angles are equal, while same-side interior angles are supplementary. Parallelism is essential to those transversal conclusions. If a diagram shows two lines without stating or establishing that they are parallel, do not import the parallel-line rules. Write the relevant equality or sum before substituting numbers so the reason for each step stays visible.
In an isosceles triangle, equal sides have equal opposite angles. In an equilateral triangle all sides agree and each angle is sixty degrees. The sum of any two side lengths must exceed the third; equivalently, if two sides are a and b, a third side c satisfies |a-b| < c < a+b. Equality would flatten the figure into a line and is excluded for a triangle. If c must be an integer, choose only integers strictly inside the interval. A largest permitted integer may be one below an integer upper bound.
A 45-45-90 triangle has sides in ratio 1:1:square root of 2, with the longest side the hypotenuse. A 30-60-90 triangle has sides in ratio 1:square root of 3:2, opposite angles thirty, sixty and ninety degrees respectively. Match sides to angles before scaling. The short leg is half the hypotenuse in the second pattern, not in every right triangle. If the angles do not establish either special pattern, use the Pythagorean theorem with the actual given lengths rather than assuming a familiar ratio.
A triangle has area one half times base times perpendicular height. A slanted side is not generally its height. A parallelogram has area base times perpendicular height, and a trapezoid has area one half times the sum of its parallel bases times their perpendicular separation. A rectangle is a special parallelogram with perpendicular adjacent sides. A rhombus has equal sides but need not be a square; its area can be computed from perpendicular diagonals as half their product. Select a formula because the necessary measurements are known, not because the shape's appearance suggests an easy number.
An n-sided simple polygon has interior-angle sum (n-2) times 180 degrees. Dividing that sum by n gives each angle only when the polygon is equiangular, as a regular polygon is. A regular polygon is both equilateral and equiangular. The exterior angles taken consistently around a convex polygon sum to 360 degrees, so each exterior angle of a regular n-gon is 360/n. Distinguish an interior angle from its supplementary exterior angle; confusing them can produce a plausible but incorrect answer.
Circumference is 2 pi r and area is pi r squared. A central angle of theta degrees selects fraction theta/360 of either the circumference for arc length or the area for sector area. Arc length and sector area use the same fraction but different whole-circle quantities. Radius is half the diameter. A tangent is perpendicular to the radius at the point of tangency. An inscribed angle subtending the same arc is half the corresponding central angle. Keep pi exact until the question requests a decimal approximation or provides a value to use. A radius drawn to the midpoint of a chord is perpendicular to that chord, creating a right triangle with half the chord as one leg. This can recover a missing chord length without assuming the chord passes through the center.
For similar figures with linear scale factor k, corresponding lengths and perimeters multiply by k while areas multiply by k squared. Doubling every side produces four times the area, not twice the area. If an area ratio is nine to twenty-five, the positive linear ratio is three to five. Congruence is the special case k = 1. Similarity must be established before transferring one side ratio to every side; equal areas alone do not imply similar shapes. A long thin rectangle can have the same area as a square without sharing its proportions.
To find the area of a uniform path outside a rectangular garden, determine the outer dimensions first. A path of width w adds w on each side of the length and on each side of the width, so both dimensions grow by 2w. Compute outer area and subtract garden area. Adding w only once to each dimension omits two strips, while multiplying the original perimeter by w omits the corner squares.
Check the model with a small sketch labeled by given lengths, but use the labels to calculate. If w were zero, the path area should become zero. Increasing w should increase the path area. The area formula can also be expanded to show a perimeter contribution plus four corner squares, offering an independent check.
This decomposition method transfers to shaded regions, frames and composite figures. Work with nonoverlapping pieces or a clear subtraction from a containing shape. Track whether a boundary measurement is an inside or outside dimension, and keep all lengths in the same units before computing square units.
A shape that looks regular may not be regular. A slanted side is not automatically a perpendicular height. Doubling a radius doubles circumference but quadruples area. A triangle inequality excludes equality. A diameter must be halved before using a radius formula. Equal area does not establish congruence or similarity.
Confirm the right-angle condition.
A right triangle has hypotenuse 13 and one leg 5.
The Pythagorean theorem is applicable.
Write the squared-length relation.
x squared + 25 = 169.
The hypotenuse is the side opposite the right angle.
Solve for the positive length.
x squared = 144, so x = 12.
A side length is positive, so discard the negative root.
Check the original relation.
25 + 144 = 169.
The computed leg is also shorter than the hypotenuse.
Read the radius and central angle.
Radius 9; central angle 120 degrees.
The question asks for sector area divided by pi.
Compute the selected fraction.
120/360 = 1/3.
The sector covers one third of the full circle.
Choose the area formula.
Whole-circle area is 81 pi.
Circumference would answer a length question instead.
Apply the fraction.
Sector area = 27 pi.
Multiply the whole area by one third.
Match the requested response.
Area divided by pi is 27.
The response asks for the coefficient, so do not enter 27 pi or its decimal approximation.
Identify corresponding sides.
Similar triangles have corresponding sides 6 and 15.
The smaller triangle's area is 20.
Calculate the linear scale factor.
k = 15/6 = 2.5.
The comparison runs from smaller to larger.
Convert to an area factor.
k squared = 6.25.
Both independent length dimensions scale by k.
Scale the known area.
20 times 6.25 = 125.
The larger triangle has area 125 square units.
Check the direction and magnitude.
The larger area exceeds twenty by more than the linear factor.
Area grows quadratically with the side scale.
State the necessary assumption.
The figures must be similar.
One pair of side measurements alone would not establish the area ratio.
Apply both triangle inequalities.
Two sides are 7 and 11, so 4 < c < 18.
The third side is an integer.
Inspect the strict endpoints.
The candidates begin at 5 and end at 17.
Determine how many integers lie in that inclusive list.
Count and verify the boundary.
A triangle has angles of 47 degrees and 68 degrees. What is its third angle in degrees?
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A right triangle has legs of lengths 9 and 12. What is its hypotenuse length?
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Two parallel lines are crossed by a transversal. One interior angle is 112 degrees. What is the interior angle on the same side of the transversal, in degrees?
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An isosceles triangle has vertex angle 38 degrees. What is either base angle in degrees?
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In a 30-60-90 right triangle, the hypotenuse is 18. What is the length of the side opposite the 30-degree angle?
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A triangle has side lengths 5, 8 and integer k. What is the greatest possible k?
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A trapezoid has parallel bases of lengths 7 and 13 and perpendicular height 6. What is its area?
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A regular octagon has how many degrees in each interior angle?
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A circle has radius 6. What is the area of a 90-degree sector divided by pi?
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Two similar triangles have corresponding side lengths 4 and 10. The smaller triangle has area 24. What is the larger triangle's area?
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A rectangular garden is 14 meters long and 10 meters wide. A path of uniform width 2 meters surrounds the garden outside its boundary. What is the area of the path in square meters?
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Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A right triangle has hypotenuse 25 and one leg 7. What is the other leg?
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Tomorrow, redraw a triangle with the same base and a different height to explain why a sketch does not fix its area. Then retrieve the special-triangle ratios and distinguish area scaling from perimeter scaling.
17. Complete an integer-side bound, step 3
17 - 5 + 1 = 13 possible values.
Four or eighteen would make a degenerate figure and must remain excluded.