45°
b
a
35
Help me if you can

9514 1404 393

  C)  a = 3√10; b = 3√5

Step-by-step explanation:

The side ratios in an isosceles right triangle are 1 : 1 : √2. This is a fact worth remembering. You know the triangle is isosceles because both acute angles are 45°, which means both legs are the same length.

If the leg lengths are 1, the Pythagorean theorem tells you the hypotenuse is ...

  c^2 = a^2 + b^2 . . . . . . for legs a, b, and hypotenuse c

  c^2 = 1^2 + 1^2 = 2

  c = √2

This tells you the leg : leg : hypotenuse ratios are as shown above.

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What that means here is that b = 3√5, the same length as the one shown. It also means that ,,,

  a = (3√5)(√2) = 3√(5·2)

  a = 3√10

The only answer choice with the correct value of 'a' is (C).

  a = 3√10, b = 3√5

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240