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In the diagram, |BC| = |BD| and |∠ABD| = 118° - Leaving Cert Mathematics - Question 4 - 2010
Question 4
In the diagram, |BC| = |BD| and |∠ABD| = 118°. (i) Find x. (ii) Find y. (b) Prove that if three parallel lines make intercepts of equal length on a transversal, ... show full transcript
Worked Solution & Example Answer:In the diagram, |BC| = |BD| and |∠ABD| = 118° - Leaving Cert Mathematics - Question 4 - 2010
Step 1
(i) Find x.
Answer
In triangle ABD,
Using the property of angles in a triangle:
ext{Sum of angles in triangle} = 180^{ ext{o}} \ \therefore |∠ABD| + |∠ACD| + |∠ADB| = 180^{ ext{o}} \ 118^{ ext{o}} + x + y = 180^{ ext{o}} \Calculating for x:
x = 180^{ ext{o}} - 118^{ ext{o}} - y \ x = 62^{ ext{o}} - y \Step 2
Step 3
Prove that if three parallel lines make intercepts of equal length on a transversal, then they will also make intercepts of equal length on any other transversal.
Answer
Given three parallel lines m, n, and r, and a transversal line intersecting them at points A, B, C respectively. By definition of similar triangles, triangles formed by these intersections will maintain proportionality.
Assuming the intercepts on the transversal are equal, we can prove:
- By triangle similarity:
- By properties of proportionality in similar triangles, corresponding segments on every transversal will also have equal lengths. Thus, this shows that equal length intercepts on one transversal imply equal lengths on any transversal.
Step 4
Step 5
(ii) Draw the image of the square under the enlargement with centre O and scale factor 2.5.
Answer
The enlargement of square OABC with a scale factor of 2.5 results in each side being multiplied:
ext{New Side} = 4 imes 2.5 = 10 ext{ cm} \The new points of the square will be A', B', C', and O'.
Step 6
(iii) Calculate the ratio area of image square : area of original square.
Answer
The area of the original square is:
ext{Area of OABC} = 4^2 = 16 ext{ cm}^2 \The area of the enlarged square is:
ext{Area of image square} = (10)^2 = 100 ext{ cm}^2 \The ratio is:
\text{Ratio} = \frac{100}{16} = 25 : 4 \Step 7
(iv) Calculate the scale factor of this enlargement.
Answer
Given the area of the square OPQR is 324 cm², its side length can be calculated as:
\text{Side length} = \sqrt{324} = 18 ext{ cm} \Using the scale factor ratio formula:
\frac{(original ext{ side})^2}{(new ext{ side})^2} = 7.5^2 : 18^2 ext{ lets denote the original side as } 7.5 \Hence, the scale factor, k, can be calculated as:
k = \frac{new ext{ side}}{original ext{ side}} = \frac{18}{7.5} = 2.4 \