Vertices of a right triangle $ABC$ with $\angle B = 90°$ are $A(3, 4)$, $B(1, 1)$ and $C(-8, 7)$. Find the value of $\tan A$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:27 · grounding rag
Model Answer
Given: ∠B = 90°, A(3, 4), B(1, 1), C(–8, 7)
Using distance formula:
$$AB = \sqrt{(3-1)^2 + (4-1)^2} = \sqrt{4+9} = \sqrt{13}$$
$$BC = \sqrt{(1-(-8))^2 + (1-7)^2} = \sqrt{81+36} = \sqrt{117} = 3\sqrt{13}$$
Since ∠B = 90°, side opposite to A is BC, and side adjacent to A is AB.
$$\tan A = \frac{\text{side opposite to } \angle A}{\text{side adjacent to } \angle A} = \frac{BC}{AB} = \frac{3\sqrt{13}}{\sqrt{13}} = 3$$
$$\therefore \tan A = 3$$
Source: Chapter 8, Section 8.1 (Trigonometric Ratios)
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Explanation
- The right angle is at B, so the hypotenuse is AC. For ∠A: opposite side = BC, adjacent side = AB.
- Use the distance formula to find BC and AB; you don't need AC here.
- tan A = BC/AB is the key ratio. Examiners expect the distance formula working shown clearly before writing the ratio.
- A common mistake is confusing opposite and adjacent sides — always identify them relative to the angle in question.