Karen Morrison and Nick Hamshaw Solutions for Exercise 7: Exercise 19.7

In the diagram, $TA$ and $TB$ are the tangents from $T$ to the circle whose centre is $O. AC$ is a diameter of the circle and $\angle ACB=x$. Find $\angle CAB$ in terms of $x$

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In the diagram, $TA$ and $TB$ are the tangents from $T$ to the circle whose centre is $O. AC$ is a diameter of the circle and $\angle ACB=x$. Find $\angle ATB$ in terms of $x$

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In the diagram, $TA$ and $TB$ are the tangents from $T$ to the circle whose centre is $O. AC$ is a diameter of the circle and $\angle ACB=x$. Find $\angle PBT$ in terms of $x$.

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The diagram shows a circular disc cut out of a square of silver sheet metal plate. The circle has a radius of $15 \mathrm{mm}$. $O$ is the centre of the circle. Calculate the length of the sides and hence the area of the uncut metal square.

The diagram shows a circular disc cut out of a square of silver sheet metal plate. The circle has a radius of $15 \mathrm{mm}$. $O$ is the centre of the circle. What area (in ${\mathrm{mm}}^2$) of metal is left over once the circle has been cut from the square? (write answer to the nearest whole number)

Mahindra makes badges by sticking an equilateral triangle onto a circular disc as shown. If the triangle has sides of $15 \mathrm{cm}$, find the diameter of the circular disc. (write answer to the nearest whole number)

The diagram shows two chords, $AC$ and $BD$, drawn in a circle. The chords intersect at the point $X$. Use angle properties to show that triangle $ABX$ is similar to triangle $CDX$.

The diagram shows two chords, $AC$ and $BD$, drawn in a circle. The chords intersect at the point $X$. Use the fact that the two triangles are similar to show that: $AX\times CX=BX\times DX$