A skewer used to test loaves in a bakery oven touches a circular baking tray, centre O, at point T, meeting the worktop at point P outside the tray. Angle T...
Question 1 Report
A skewer used to test loaves in a bakery oven touches a circular baking tray, centre O, at point T, meeting the worktop at point P outside the tray. Angle TPO is \(x^{\circ}\) and angle TOP is \((2x + 15)^{\circ}\).
- Angle OTP \(= 90^{\circ}\); use this to write an equation in \(x\) from the triangle angle sum, and solve it. (2)
- Work out the size of angle TOP. (1)
Answer Details
A tangent is always perpendicular to the radius at the point where it touches the circle, which fixes one angle of triangle OTP so the triangle's angle sum gives an equation in \(x\).
(a) With angle OTP \(=90^\circ\) (tangent perpendicular to radius), the angles of triangle OTP sum to \(180^\circ\): \( 90+x+(2x+15)=180 \) [1 mark]. Combining like terms gives \( 3x+105=180 \), so subtracting 105 and dividing by 3 gives \( x=25 \) [1 mark].
(b) Substituting \( x=25 \): angle TOP \(=2(25)+15=65^\circ\) [1 mark].
As a check, all three angles of triangle OTP now sum correctly: \( 90+25+65=180 \), confirming both the right angle assumption and the value of \(x\).
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