Mathematics

Problem

In right-angled triangle

XYZ

with

\angle Y = 90^{\circ}

, the lengths of the two sides adjacent to the right angle are

XY = 3k + 2\,\text{cm}

and

YZ = 4k - 1\,\text{cm}

, where

k

is a real number greater than

1

. a. (i) Express the length of the hypotenuse

XZ

in terms of

k

. b. (ii) For

\angle X

, obtain simplified expressions for

\sin X

and

\cos X

in terms of

k

. c. (iii) Prove that

\sin^{2} X + \cos^{2} X = 1

for all permissible values of

k

. d. (iv) If

k = 3

, calculate the numerical values of

\tan X

and

\csc X

. (Leave answers as simplified fractions where possible.)

Intuitive explanation

The key is recognizing that the hypotenuse $XZ$ must satisfy the Pythagorean theorem, and any constraint on the triangle (like a specific perimeter, area, or hypotenuse length) will give you an equation to solve for $k$ . Once you set up that equation using $XZ^2 = (3k+2)^2 + (4k-1)^2$ , you'll have a quadratic in $k$ that determines the actual dimensions.

Intuitive explanation

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