If c is the measure of the hypotenuse of a right triangle, how do you find each missing measure given a=x, b=x+41, c=85?

Question

Eva Abramson · Accepted Answer

36 & 77

We know, In a right triangle, #hypotenuse^2 = base^2+height^2# Here, #c^2 = a^2 + b^2# #rArr 85^2 = x^2 + (x+41)^2# #rArr 7225 = x^2+x^2+82x+1681# #rArr 2x^2+82x+1681-7225=0# #rArr# 2x^2 + 82x - 5544=0# #rArr 2(x^2+41x-2772)=0# #rArr x^2+41x-2772 = 0# #rArr x^2+77x-36x-2772 = 0# #rArr x(x+77)-36(x+77)=0# #rArr (x-36)(x+77)=0# #rArr x = 36 & -77# #[x != -77]# Hence a = 36 & b = 36+41 = 77

Jace Anderson · Answer

undefined To find the missing measures of the right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

Using the given values, we have:
c = 85
a = x
b = x + 41

Applying the Pythagorean theorem, we can set up the equation:
c^2 = a^2 + b^2

Substituting the given values:
85^2 = x^2 + (x + 41)^2

Simplifying the equation:
7225 = x^2 + (x^2 + 82x + 1681)

Combining like terms:
7225 = 2x^2 + 82x + 1681

Rearranging the equation to standard form:
2x^2 + 82x + 1681 - 7225 = 0

Simplifying further:
2x^2 + 82x - 5544 = 0

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Once we find the value of x, we can substitute it back into the expressions for a and b to find their respective measures.