Right-Angled Triangles (Pythagoras & Trigonometry) (Cambridge International IGCSE Mathematics): Exam Questions

The diagram shows a quadrilateral, $ABCD$, formed from two triangles, $ABC$ and $ACD$.

$ABC$ is a right-angled triangle.

Calculate angle $BAC$.

Angle $BAC$ = ..............................................

Extract:

The diagram shows a quadrilateral, A B C D, formed from two triangles, A B C and A C D. A B C is a right-angled triangle.

Calculate the shortest distance from $D$ to $AC$.

.......................................... cm

The diagram shows a right-angled triangle.

Calculate the perimeter.

............................................ cm

The diagram shows a right-angled triangle.

Calculate the value of $x$.

$x$ = ............................................

Calculate the length of AC.

*AC *= ........................................ cm

Calculate AC.

AC = .................................... m

Here is a right-angled triangle.

Work out the length of $AC$. Give your answer correct to 1 decimal place.

$LMN$is a right-angled triangle.

$MN$= 9.6 cm.

$LM$= 6.4 cm.

Calculate the size of the angle marked $x^{\circ}$.

Give your answer correct to 1 decimal place.

$PQR$is a right-angled triangle.

$PR$ = 17 cm

$PQ$= 25 cm

Work out the size of angle $RPQ$.

Give your answer correct to 1 decimal place.

$ABC$is a right-angled triangle.

Calculate the length of $AB$.

Give your answer correct to 3 significant figures.

$PQR$is a right-angled triangle.

Work out the size of the angle marked $x$.

Give your answer correct to 1 decimal place.

*ABDF *is a parallelogram and *BCDE *is a straight line.

AF = 12cm, *AB *= 9 cm, angle *CFD *= 40° and angle FDE = 80°.

Calculate the height, $h$, of the parallelogram.

$h=$ ........................................... cm

The diagram shows a prism with length 18 cm and volume 253.8 cm³.

The cross-section of the prism is a right-angled triangle with base 6 cm and height $h$ cm.

Show that the value of $h$ is 4.7 .

Calculate the value of $x$.

$x$ = ...............................................

Extract:

The diagram shows a prism with length 18 cm and volume 253.8 cm^3^. The cross-section of the prism is a right-angled triangle with base 6 cm and height 4.7 cm.

Calculate the total surface area of the prism.

........................................ cm²

The diagram shows a circle, centre O. The straight line ABC is a tangent to the circle at B. OB = 8 cm, AB = 15 cm and BC = 22.4 cm. AO crosses the circle at X and OC crosses the circle at Y.

Calculate angle XOY.

Angle XOY = .............................................

In the diagram, AB and CD are parallel. AD and BC intersect at right angles at the point X. AB = 10 cm, CD = 5 cm, AX = 8 cm and BX = 6 cm.

Calculate angle XAB.

Angle XAB = ..............................................

Here is part of a field.

This part of the field is in the shape of a trapezium.

A farmer wants to put a fence all the way around the edge of this part of the field.

The farmer has 50m of fence.

Does he have enough fence? You must show all your working.

The diagram shows a rectangular framework.

The framework is made from 5 metal rods.

The metal rods have a weight of 0.9 kg per metre.

Work out the total weight of the framework.

Give your answer, in kg, correct to 3 significant figures.

The diagram shows a rectangle and a diagonal of the rectangle.

Work out the length of the diagonal of the rectangle.

Give your answer correct to 1 decimal place.

.....................$\mathrm{cm}$

Calculate the value of the length $YZ$.

Find the interior angle at $X$.

Extract:

The diagram shows a quadrilateral, [A B C D], formed from two triangles, [A B C] and [A C D].

Angle BAC is 37°. AB=18 cm.

The diagram shows a quadrilateral, $ABCD$, formed from two triangles, $ABC$ and $ACD$.

Calculate the shortest distance from $B$ to $AC$.

.......................................... cm

The diagram shows a circle, centre O.

AB is a chord of length 12 cm.

M is the mid-point of AB and OM = 4.5 cm.

Calculate the radius of the circle.

.......................................... cm

$ABC$is an isosceles triangle.

Work out the area of the triangle.

Give your answer correct to 3 significant figures.

Calculate the length of $PR$.

Give your answer correct to 3 significant figures.

$ABC$is a right-angled triangle.

Work out the size of angle $ABC$.

Give your answer correct to 1 decimal place.

Extract:

[A B C] is a right-angled triangle. Angle [A C B] is still 90°. The length of the side [B C] is still 7 cm.

The length of the side $AB$is reduced by 1 cm.

The length of the side $BC$is still 7 cm.

Angle $ACB$is still 90°

Will the value of cos$ABC$increase or decrease?

You must give a reason for your answer.

A solid metal cone has radius 1.65 cm and slant height 4.70 cm.

Find the angle the slant height makes with the base of the cone.

The diagram shows a trapezium $ABCD$.

$AB$ is parallel to $DC$.

$AB=55m$, $BD=70m$, angle $ABD=40^{\circ}$ and angle $BCD=32^{\circ}$.

Calculate the shortest distance from $A$ to $BD$.

............................................ m

The straight line $AC$* *has equation $y=4x+5$.

Calculate the acute angle between $AC$ and the $x$-axis.

In this question, all measurements are in metres.

The diagram shows a right-angled triangle.

Given that $x=3.82$ to 3 significant figures, calculate the smallest angle of the triangle.

In the pentagon $ABCDE$, angle $ACB$ = angle $AED$ = 90°. Triangle $ACD$ is equilateral with side length 12 cm.

$DE$ = $BC$ = 6 cm.

Calculate angle $BAE$.

Angle $BAE$ = ...............................................

Extract:

In the pentagon A B C D E, angle A C B = angle A E D = 90°. Triangle A C D is equilateral with side length 12 cm. D E = B C = 6 cm.

Calculate $AB$.

$AB$ = ......................................... cm

Extract:

In the pentagon [A B C D E], angle [A C B] = angle [A E D] = 90°. Triangle [A C D] is equilateral with side length 12 cm.

[D E] = [B C] = 6 cm.

Calculate $AE$.

$AE$ = ......................................... cm

The cross-section of a prism is a right-angled isosceles triangle with its equal sides each being 7 cm.

The length of the prism is 18 cm.

Calculate the total surface area of the prism.

......................................... cm²

$C$ lies on a circle with diameter $AD$.

$B$ lies on $AC$ and $E$ lies on $AD$ such that $BE$ is parallel to $CD$.

$AB$ = 21 cm, $CD$ = 18 cm and $BE$ = 13.5 cm.

Work out the radius of the circle.

............................................ cm

Point A has co-ordinates (1, 0) and point B has co-ordinates (2, 5).

Calculate the angle between the line AB and the $x$-axis.

A square, with sides of length $x$ cm, is inside a circle. Each vertex of the square is on the circumference of the circle.

The area of the circle is 49 cm².

Work out the value of $x$.

Give your answer correct to 3 significant figures.

Here is triangle $ABD.$

The point $C$ lies on $BD$.

$AD=13\mathrm{cm}BC=8\mathrm{cm} \mathrm{angle}ADB=90^{\circ}\mathrm{angle}CAD=20^{\circ}$

Calculate the size of angle $BAC$.

Give your answer correct to $1$ decimal place.

The diagram shows the positions of three points $A$, $B$ and $C$ in a field.

Calculate the shortest distance from point $B$ to $AC$.

............................................. m

The diagram shows a field ABCD.

The bearing of B from A is 140°.

C is due east of B and D is due north of C.

AB = 400m, BC = 350m and CD = 450m.

Calculate the distance from D to A.

............................................. m

The angle $BAC$* is $82.8^{\circ}$ *correct to 3 significant figures.

D is the point on AC that is nearest to B.

Calculate the distance from D to A.

................................................ m

The diagram shows a field ABCDE.

Calculate the perimeter of the field ABCDE.

................................................ m

In the diagram, $BC$ is a vertical wall standing on horizontal ground $AB$. $AC$ = 76.7 m.

$D$ is the point on $AB$ where $AD$ = 58 m.

The angle of elevation of $C$ from $A$ is 26°.

The angle of elevation of $C$ from $D$ is 72°.

Calculate $BD$.

$BD$ = .......................................... m