Question

1. what is the length of each hypotenuse? give your answer to the nearest tenth of a centimetre. a) y=7 cm x=6 cm z b) c=11 cm b=8 cm d

Explanation:

Part (a)

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem, which states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\). Here, \(x = 6\space cm\) and \(y=7\space cm\) are the legs, and \(z\) is the hypotenuse.

Step2: Apply the Pythagorean theorem

\(z^{2}=x^{2}+y^{2}\)
Substitute \(x = 6\) and \(y = 7\) into the formula:
\(z^{2}=6^{2}+7^{2}=36 + 49=85\)

Step3: Solve for \(z\)

Take the square root of both sides: \(z=\sqrt{85}\approx9.2\space cm\) (rounded to the nearest tenth)

Part (b)

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem. Here, \(c = 11\space cm\) and \(b = 8\space cm\) are the legs, and \(d\) is the hypotenuse.

Step2: Apply the Pythagorean theorem

\(d^{2}=c^{2}+b^{2}\)
Substitute \(c = 11\) and \(b = 8\) into the formula:
\(d^{2}=11^{2}+8^{2}=121 + 64 = 185\)

Step3: Solve for \(d\)

Take the square root of both sides: \(d=\sqrt{185}\approx13.6\space cm\) (rounded to the nearest tenth)

Answer:

a) The length of \(z\) is approximately \(\boldsymbol{9.2}\space cm\).
b) The length of \(d\) is approximately \(\boldsymbol{13.6}\space cm\).