Find Side (h) Of Given Triangle | Radius of InCircle is 2- Important Geometry Skills Explained

Find Side (h) Of Given Triangle | Radius of InCircle is 2- Important Geometry Skills Explained

Learn how to find altitude H of the given triangle. One leg measures 10 units. Radius of Incircle is 2 units. Important Geometry and algebra skills are also explained: Area of right triangle, Tangents drawn from an exterior point to a Circle are equal, Line segment joining centre and point of tangency is perpendicular to the tangent. Step-by-step tutorial by @MathbookYouTube

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Find Altitude H Of Triangle | Radius of InCircle is 2- Important Geometry & Algebra Skills Explained
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[jimlocke]

Method using trigonometry: Designate the center of the circle as point O. We construct OB and OR to form ΔOPB and ΔORB. Let <OBP = Θ. We note that tan(Θ)= 2/8 = 1/4. ΔOPB and ΔORB are congruent by side angle side. By congruency, <OBP = <OBR. Furthermore, <ABC = <OBP + <OBR = Θ + Θ = 2Θ. We use the formula for tangent of sum of angles: tan(a + b) = (tan(a) + tan(b))(1 - ((tan(a))(tan(b))). In this case, the 2 angles are equal. So, the formula becomes tan(<ABC) = tan(2Θ) = (tan(Θ) + tan(Θ))/(1 - ((tan(Θ))(tan(Θ))) = (1/4 + 1/4)/(1 - ((1/4)(1/4))) = (1/2)/(15/16) = 16/30 = 8/15. However, tan(<ABC) = h/AB = h/10 and tan(<ABC) = tan(2Θ) = 8/15. So h/10 = 8/15 and h = 80/15 = 16/3, as MathBook also found.

Checking our work (as well as MathBook's): By the Pythagorean theorem, length AC = √(h² + 10²) = √((16/3)² + 10²) = √(256/9) + 100) = √(1156/9) = 34/3 which must equal h - 2 + 8 = (16/3) + 6 = 34/3. Our calculation checks out.