Question

1. in trapezoid abcd, calculate the length of diagonal ac to the nearest tenth. show your work. (3 marks)

image of trapezoid abcd with labeled sides: ab = 6.6 cm, bc = 10.2 cm, cd = 6.6 cm, ad = 15.8 cm, and a right angle at d for triangle c d (part of the trapezoid)

Explanation:

Step1: Find base of right triangle

First, calculate the length of the base of the right triangle formed by dropping the height from C to AD.
$15.8 - 10.2 = 5.6$ cm

Step2: Calculate height of trapezoid

Use the Pythagorean theorem on the right triangle with hypotenuse 6.6 cm and base 5.6 cm to find height $h$.
$h = \sqrt{6.6^2 - 5.6^2} = \sqrt{43.56 - 31.36} = \sqrt{12.2} \approx 3.493$ cm

Step3: Find base for triangle ACD

The base of triangle ACD (along AD) is 15.8 cm, and height is $h \approx 3.493$ cm. Wait, alternatively, use the horizontal distance from A to the foot of the height: $15.8 - 5.6 = 10.2$ cm. Now use Pythagorean theorem for diagonal AC:
$AC = \sqrt{10.2^2 + 3.493^2}$
First compute $10.2^2 = 104.04$, $3.493^2 \approx 12.2$
$AC = \sqrt{104.04 + 12.2} = \sqrt{116.24} \approx 10.8$ cm

Answer:

10.8 cm