Question

Given:

Answer 1

Answer:

$PS=13\text{ units}$

Step-by-step explanation:

So, we know that PR is 20, SR is 11, and QS is 5.

We also know that PQ is perpendicular to QR, forming the right angle at ∠Q.

We know all the side lengths except for PQ and PS (the one we want to find). Notice that if we find PQ first, we can then use the Pythagorean Theorem to find PS since we already know QS.

So, let's find PQ.

We can see that we can also use the Pythagorean Theorem on PQ. PQ, QR, and PR (the hypotenuse) will be our sides. So:

$(PQ)^2+(QR)^2=(PR)^2$

We know that PR is 20.

QR is the combined length of QS+SR, so QR is 5+11 or 16.

So, substitute:

$(PQ)^2+(16)^2=(20)^2$

Solve for PQ. Square:

$(PQ)^2+256=400$

Subtract 256 from both sides:

$(PQ)^2=144$

Take the square root of both sides:

$PQ=12$

So, the side length of PQ is 12.

Now, we can use the Pythagorean Theorem again to find PS. Notice that PQ, QS, and PS also form a right triangle, with PS being the hypotenuse. So:

$(PQ)^2+(QS)^2=(PS)^2$

We already know that QS is 5. We also just determined that PQ is 12. Substitute:

$(12)^2+(5)^2=(PS)^2$

Square:

$144+25=(PS)^2$

Add:

$169=(PS)^2$

Take the square root of both sides:

$PS=13$

Therefore, the length of PS is 13 units.

And we're done!