Question

if pqr is a equilateral triangle and PX perpendicular at QR,then PX^2 is equal to ​

Answer 1

Answer:

PX² = 3PR²/4 which by using substitution could also be written as

PX² = 3RX²

Step-by-step explanation:

Equilateral triangle mean all side equal so we know that PQ = QR = PR

If PX is perpendicular at QR that mean the line is forming a right angle which means that X lies half way between triangle side QR.  So QX = XR = QR/2

We can then use the Pythagoras Theorem on the right angle triangle of PXR   The hypotenuse would be PR, so:

PX²+XR² = PR²

We are solving for PX², so we subtract RX² from both sides.

PX² = PR²-XR²

We know that XR can also be written as QR/2, so

PX² = PR² - (QR/2)²

We now have PR and QR on one side, but we also know that PR = QR, so

PX² = PR² - (PR/2)²

PX² = PR² - PR²/4

We now need to have a common denominator in order to subtract

PX² = (4PR²/4) - (PR²/4) which also can be written as (4PR²- PR²)/4

PX² = 3PR²/4

So PX² = 3PR²/4 which can also be written as 3/4 × PR²

Depending on the assignment, they may have different answers available.

Since PR = QR and QR is RX × 2

PX² = (3/4) × (RX × 2)²

PX² = (3/4) × RX² × 4, since 4 × 3/4 = 3

PX² = 3RX²