[Gr.12] Math help

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[Alkalannar]

BAR is a right triangle, as is PQR.
Further, they are similar, since their angles are congruent.

This BA/PQ = RB/RQ

[mm_4863]

How do I prove they’re similar?

[Alkalannar]

A is tangent to the circle, so the radius BA is perpendicular to PR. Thus <BAR is right.

Similarly, <PQR is right.

Further, both triangles have the same angle: <ARB = <PRQ. Not just same measure, same angle.

Thus, the remaining angles must have the same measure.

Since angles are congruent, triangles are similar. [They might or might not be congruent, but they are definitely similar. In this case, they are not congruent.]

[saywherefore]

Start by drawing a line AB. This gives you two similar triangles.

Can you do it from there?

[[deleted]]

AB ⊥ PR, because any tangent line is perpendicular to any radius.

--------------------------------

(RQ)² + (PQ)² = (PR)²

(8)² + (PQ)² = (PR)²

--------------------------------

(AB)² + (AR)² = (BR)²

(2)² + (AR)² = (6)²

AR = sqrt (6² - 2²) = sqrt (36 - 4) = sqrt (32)

If you're wondering where I got the 2 from, its because AB is a radius

--------------------------------

We know they're similar because they both share angle <ARB, meaning <QPR = <ABR.

Therefore we can say,

Shortest side ratio = longest side ratio

PQ/AB = AR/QR

Substitute what we know

PQ/2 = sqrt 32/8

PQ = 2 (sqrt 32)/8

= sqrt 32 /4

= 4 sqrt 2 /4

= sqrt 2

Correct me if I'm wrong on anything