Question

a cartographer is attempting to find the distance across a small lagoon. starting at point a on the south side of the lagoon he walks 30 yards to point c, and then makes a 90° turn and walks another 72 yards before coming to point b, directly across the lagoon from point a. what is the distance from point a to point b?

Explanation:

Step1: Identify the right - angled triangle

We have a right - angled triangle $\triangle ABC$ with legs $AC = 30$ yards and $BC=72$ yards, and we want to find the hypotenuse $AB$.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - angled triangle with legs $a$ and $b$ and hypotenuse $c$, $c^{2}=a^{2}+b^{2}$. Here, $a = 30$, $b = 72$, and $c=AB$. So, $AB^{2}=30^{2}+72^{2}$.
$AB^{2}=900 + 5184$.
$AB^{2}=6084$.

Step3: Find the value of $AB$

Take the square root of both sides: $AB=\sqrt{6084}=78$ yards.

Answer:

78 yards