Applying Geometric Theorems Flashcards

A 15-foot ladder is leaning against a wall. The base of the ladder is 9 feet away from the base of the wall. How high up the wall does the ladder reach?

Answer: 12 feet

This scenario forms a right triangle, with the ladder as the hypotenuse. Using the Pythagorean theorem (a² + b² = c²), we have a² + 9² = 15². This simplifies to a² + 81 = 225. Subtracting 81 gives a² = 144. Taking the square root, a = 12 feet.

A triangle has side lengths of 8 cm, 15 cm, and 17 cm. Which statement is true about this triangle?

Answer: It is a right triangle.

To determine if it's a right triangle, use the converse of the Pythagorean theorem: a² + b² = c². Check if 8² + 15² = 17². This becomes 64 + 225 = 289, which simplifies to 289 = 289. Since the equation is true, the triangle is a right triangle.

In two triangles, two pairs of corresponding angles and the pair of corresponding non-included sides are congruent. Which theorem proves the triangles are congruent?

Answer: Angle-Angle-Side (AAS)

The Angle-Angle-Side (AAS) congruence theorem states that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the triangles are congruent. The description matches this theorem perfectly.

Given that ΔABC ≅ ΔXYZ, AB = 2x + 1, and XY = 15. What is the value of x?

Answer: x = 7

If two triangles are congruent, their corresponding parts are congruent (CPCTC). Therefore, side AB corresponds to side XY, and their lengths must be equal. Set up the equation 2x + 1 = 15. Solving for x gives 2x = 14, so x = 7.

A 6-foot tall person casts a 4-foot shadow. At the same time, a nearby flagpole casts a 20-foot shadow. How tall is the flagpole?

Answer: 30 feet

The person and their shadow, and the flagpole and its shadow, form two similar right triangles. Set up a proportion of corresponding sides: (person's height) / (person's shadow) = (flagpole's height) / (flagpole's shadow). This gives 6/4 = h/20. Cross-multiply to get 4h = 120, so h = 30 feet.

Two triangles are similar. The sides of the first triangle are 5, 12, and 13. The shortest side of the second triangle is 15. What is the perimeter of the second triangle?

Answer: 90

First, find the scale factor by comparing the corresponding shortest sides: 15 / 5 = 3. This means all sides of the second triangle are 3 times larger. The other sides are 12*3=36 and 13*3=39. The perimeter of the second triangle is 15 + 36 + 39 = 90. Alternatively, find the perimeter of the first triangle (5+12+13=30) and multiply by the scale factor (30*3=90).

In ΔPQR, ∠Q is a right angle. Point S is on PR such that QS is perpendicular to PR. Which statement proves that ΔPQS ~ ΔQRS?

Answer: Angle-Angle (AA) Similarity

In a right triangle, the altitude to the hypotenuse creates three similar triangles. Both smaller triangles (ΔPQS and ΔQRS) are similar to the large triangle (ΔPQR) and to each other. This similarity is proven by Angle-Angle (AA) similarity, as each small triangle shares an angle with the large triangle and both have a right angle.