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Calculate Angle x in a Right Triangle with Given Triangle Area and Segment Lengths
Mathematics
Grade 9 (Junior High School)
Question Content
Points B, C and D lie on a straight line. AC is perpendicular to BD. BC = 6 cm and BD = 22 cm. Triangle ABC has area 42 cm². Calculate the size of angle x. You must show all your working.
Correct Answer
35° (rounded to the nearest whole number, or 34.99° for precise value)
Detailed Solution Steps
1
Step 1: Calculate the length of AC using the area of triangle ABC. The formula for the area of a triangle is $Area = \\frac{1}{2} \\times base \\times height$. For triangle ABC, base = BC = 6 cm, area = 42 cm², and height = AC. Substitute into the formula: $42 = \\frac{1}{2} \\times 6 \\times AC$. Solve for AC: $AC = \\frac{42 \\times 2}{6} = 14$ cm.
2
Step 2: Calculate the length of CD. Since B, C, D are collinear, $BD = BC + CD$. We know BD = 22 cm and BC = 6 cm, so $CD = BD - BC = 22 - 6 = 16$ cm.
3
Step 3: Use right triangle trigonometry to find angle x. In right triangle ACD, AC = 14 cm (opposite side to angle x), CD = 16 cm (adjacent side to angle x). The tangent function is $\\tan(x) = \\frac{opposite}{adjacent} = \\frac{AC}{CD} = \\frac{14}{16} = 0.875$.
4
Step 4: Calculate angle x using the inverse tangent function: $x = \\tan^{-1}(0.875) ≈ 34.99°$, which rounds to 35°.
Knowledge Points Involved
1
Area of a triangle
The formula for the area of a triangle is $Area = \\frac{1}{2} \\times base \\times perpendicular height$. It applies to all triangles, and is used here to find an unknown side length when the area and one side are given.
2
Collinear points segment addition
If three points lie on a straight line (collinear), the total length of the segment between the two outer points is equal to the sum of the lengths of the two smaller segments between the inner point and each outer point. Here, $BD = BC + CD$.
3
Right triangle trigonometry (tangent function)
For an acute angle in a right triangle, the tangent function is defined as $\\tan(\\theta) = \\frac{length of opposite side}{length of adjacent side}$. It is used to relate the sides of a right triangle to its acute angles, and the inverse tangent function $\\tan^{-1}$ is used to find the angle when the opposite and adjacent sides are known.
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