The diagram shows a triangle, ABC, with perpendicular height BD - OCR - GCSE Maths - Question 14 - 2023 - Paper 5
Question 14
The diagram shows a triangle, ABC, with perpendicular height BD.
BC = 12 cm, angle BCD = 30° and angle BAD = 45°.
Work out the length of BD.
(a) ...................... show full transcript
Worked Solution & Example Answer:The diagram shows a triangle, ABC, with perpendicular height BD - OCR - GCSE Maths - Question 14 - 2023 - Paper 5
Step 1
Work out the length of BD.
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Answer
To find the length of BD, we can use trigonometric ratios.
First, identify angle BDA, which is 45ext°. Since triangle ABD is a right triangle, we can apply the tangent function:
tan(45ext°)=ADBD
From this, we know that AD=BD since tan(45ext°)=1.
Now, move on to triangle BCD, where we know angle BCD is 30ext° and BC = 12 cm. Using the sine function, we have:
sin(30ext°)=BCBD
Thus:
21=12BD
Therefore, we can rearrange it to find BD:
BD=12×21=6extcm.
So, the length of BD is 6 cm.
Step 2
Work out the exact length of AB.
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Answer
To determine the length of AB, we apply the sine rule or cosine rule in triangle ABD. Since angle BAD is 45ext° and angle BDA is also 45ext°, we can calculate:
In triangle ABD, we state:
AB=sin(30ext°)BD=216=12extcm.
