AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Find x and y in a 45-45-90 Right Isosceles Triangle with Leg Length 8
Mathematics
Grade 8 (Junior High School)
Question Content
In a right isosceles triangle with one leg length 8 and a 45° acute angle, find the values of x (the other leg) and y (the hypotenuse).
Correct Answer
x = 8, y = 8√2
Detailed Solution Steps
1
Step 1: Identify the triangle type. This is a 45-45-90 right triangle (right isosceles triangle), which has two equal legs and a hypotenuse that is √2 times the length of a leg.
2
Step 2: Solve for x. In a 45-45-90 triangle, the two legs are congruent. Since one leg is 8, the other leg x = 8.
3
Step 3: Solve for y. Use the 45-45-90 triangle side ratio: hypotenuse = leg × √2. Substitute the leg length 8, so y = 8 × √2 = 8√2.
Knowledge Points Involved
1
45-45-90 Special Right Triangle Properties
A 45-45-90 triangle is a right isosceles triangle, meaning it has two equal acute angles (45° each) and two congruent legs. The ratio of its sides is leg : leg : hypotenuse = 1 : 1 : √2. This ratio allows quick calculation of unknown side lengths if one side is known, without needing to use the Pythagorean theorem or trigonometric functions every time.
2
Congruent Sides in Isosceles Triangles
In any isosceles triangle, the sides opposite equal angles are congruent. In a 45-45-90 right triangle, the two 45° angles are equal, so the legs opposite these angles are equal in length.
3
Pythagorean Theorem (Verification)
The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are legs, and c is the hypotenuse. For this triangle, 8² + 8² = y² → 64 + 64 = y² → 128 = y² → y = √128 = 8√2, which confirms the special triangle ratio result.
Loading solution...
