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Triangles and the Pythagorean theorem — SHSAT practice questions

On the SHSAT, Triangles and the Pythagorean theorem covers right and isosceles triangles, including side lengths, angles, area, and perimeter. Students must apply the Pythagorean theorem to algebraic legs and hypotenuses, find each base angle from a vertex angle, and recover missing sides from a given hypotenuse and area. Items usually ask for a perimeter or an angle measure rather than a named theorem. Common traps include swapping a leg with the hypotenuse, treating an isosceles triangle as equilateral, and stopping after one side instead of finishing the perimeter. Answer choices often look like a single side, a doubled angle, or a half-finished perimeter.

  • Section: Mathematics 57 questions on the paper
  • 93 practice questions in the app

Sample questions

Question 1

A right triangle has legs measuring xx and x+4x+4 units and hypotenuse measuring x+8x+8 units. What is the perimeter of the triangle?

  1. A44
  2. B48
  3. C52
  4. D60
Show answer

Answer: B — 48

x2+(x+4)2=(x+8)2x^2+(x+4)^2=(x+8)^2 simplifies to x2−8x−48=0x^2-8x-48=0, so x=12x=12. The side lengths are 12, 16, and 20, and the perimeter is 48.

Question 2

An isosceles triangle has a vertex angle of 40∘40^\circ. What is the measure of each base angle?

  1. A40∘40^\circ
  2. B50∘50^\circ
  3. C70∘70^\circ
  4. D80∘80^\circ
Show answer

Answer: C — 70∘70^\circ

The base angles sum to 180−40=140180-40=140 degrees, so each base angle measures 140÷2=70140\div2=70 degrees.

Question 3

A right triangle has a hypotenuse of length 20 units and an area of 96 square units. What is the perimeter of the triangle?

  1. A40
  2. B44
  3. C48
  4. D52
Show answer

Answer: C — 48

If the legs are aa and bb, then ab=192ab=192 and a2+b2=400a^2+b^2=400. Therefore, (a+b)2=400+2(192)=784(a+b)^2=400+2(192)=784, so a+b=28a+b=28 and the perimeter is 48.

Practice 93 Triangles and the Pythagorean theorem questions in the app

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