The diagram shows a trapezium $ABCD$, where $A$ is $(-12,\ -2)$, $B$ is $(-10,\ b)$ a

Question

The diagram shows a trapezium $ABCD$, where $A$ is $(-12,\ -2)$, $B$ is $(-10,\ b)$ and $C$ is $(4,\ 6)$. The point $D(0,\ k)$ lies on the $y$-axis such that $AD$ is parallel to $BC$ and the $y$-axis bisects angle $ADC$. Express the gradients of $AD$ and $CD$ in terms of $k$.[2] Hence show that $k=10$.[2] Find the value of $b$.[2] Show that angle $ABC$ is not $90^\circ$.[2] Find the area of the trapezium $ABCD$.[3]

Accepted Answer

(i) $m_{AD}=\dfrac{k+2}{12}$, $m_{CD}=\dfrac{6-k}{4}$ (ii) $k=10$ (iii) $b=-8$ (iv) $m_{AB}m_{BC}=-3\neq-1$, so $\angle ABC\neq90^\circ$ (v) $104\text{ units}^2$

Trapezium with the y-axis bisecting an angle

Trapezium with the y-axis bisecting an angle

Solution:

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