The town of Numan is 104 km from Jalingo
on a bearing of North East. Yola is 56 km
from Numan on a bearing of South .
Sketch a diagram showing the points of the
three towns. Calculate:
(a) the distance of Yola from Jalingo
(b) the bearing of Yola from Jalingo.

Ask by Hall Park. in Nigeria

Mar 23,2025

Question

The town of Numan is 104 km from Jalingo
on a bearing of North East. Yola is 56 km
from Numan on a bearing of South .
Sketch a diagram showing the points of the
three towns. Calculate:
(a) the distance of Yola from Jalingo
(b) the bearing of Yola from Jalingo.

Ask by Hall Park. in Nigeria

Mar 23,2025

UpStudy AI · Accepted Answer

We set up a coordinate system with Jalingo at the origin, the positive $$ y $$–axis pointing north and the positive $$ x $$–axis pointing east.

**Step 1. Locate Numan**

Numan is 104 km from Jalingo on a bearing of North $$ 50^\circ $$ East. In our coordinate system this gives:
$$
x_{\text{Numan}} = 104\sin(50^\circ), \quad y_{\text{Numan}} = 104\cos(50^\circ).
$$

**Step 2. Locate Yola relative to Numan**

Yola is 56 km from Numan on a bearing of South $$ 60^\circ $$ East. A "South $$ 60^\circ $$ East" direction means that starting from due south the line is turned $$ 60^\circ $$ toward the east. Hence the displacement from Numan is:

$$
x_{\text{displacement}} = 56\sin(60^\circ), \quad y_{\text{displacement}} = -56\cos(60^\circ).
$$
(The $$ y $$–component is negative since the motion is toward the south.)

Thus the coordinates of Yola (relative to Jalingo) are:
$$
x_{\text{Yola}} = 104\sin(50^\circ) + 56\sin(60^\circ),
$$
$$
y_{\text{Yola}} = 104\cos(50^\circ) - 56\cos(60^\circ).
$$

**Step 3. Numerical Calculation**

Using approximate trigonometric values:
$$
\sin(50^\circ) \approx 0.7660,\quad \cos(50^\circ) \approx 0.6428,
$$
$$
\sin(60^\circ) \approx 0.8660,\quad \cos(60^\circ) = 0.5.
$$

We compute:
$$
x_{\text{Numan}} \approx 104 \times 0.7660 = 79.66,
$$
$$
y_{\text{Numan}} \approx 104 \times 0.6428 = 66.77.
$$

The displacement from Numan:
$$
x_{\text{disp}} \approx 56 \times 0.8660 = 48.50,
$$
$$
y_{\text{disp}} \approx -56 \times 0.5 = -28.
$$

Now, Yola’s coordinates are:
$$
x_{\text{Yola}} \approx 79.66 + 48.50 = 128.16,
$$
$$
y_{\text{Yola}} \approx 66.77 - 28 = 38.77.
$$

**Step 4. Distance of Yola from Jalingo (Part a)**

The distance $$ d $$ is given by:
$$
d = \sqrt{x_{\text{Yola}}^2 + y_{\text{Yola}}^2} \approx \sqrt{(128.16)^2 + (38.77)^2}.
$$

Calculate:
$$
(128.16)^2 \approx 16419.5, \quad (38.77)^2 \approx 1503.5,
$$
$$
d \approx \sqrt{16419.5 + 1503.5} = \sqrt{17923} \approx 133.96\,\text{km}.
$$

Thus, Yola is approximately $$ 134\,\text{km} $$ from Jalingo.

**Step 5. Bearing of Yola from Jalingo (Part b)**

The bearing is measured from the north direction (the positive $$ y $$–axis). We first calculate the angle $$ \theta $$ that the line joining Jalingo to Yola makes with the north. With the east–component $$ x_{\text{Yola}} $$ and the north–component $$ y_{\text{Yola}} $$, we have:
$$
\theta = \arctan\left(\frac{x_{\text{Yola}}}{y_{\text{Yola}}}\right) \approx \arctan\left(\frac{128.16}{38.77}\right).
$$

Compute the ratio:
$$
\frac{128.16}{38.77} \approx 3.305,
$$
so
$$
\theta \approx \arctan(3.305) \approx 73^\circ.
$$

Thus, the bearing is $$ \text{North } 73^\circ \text{ East} $$.

**Final Answers**

(a) The distance of Yola from Jalingo is approximately $$ 134\,\text{km} $$.

(b) The bearing of Yola from Jalingo is $$ \text{North } 73^\circ \text{ East} $$.
Yola is approximately 134 km from Jalingo on a bearing of North 73° East.

The town of Numan is 104 km from Jalingo
on a bearing of North East. Yola is 56 km
from Numan on a bearing of South .
Sketch a diagram showing the points of the
three towns. Calculate:
(a) the distance of Yola from Jalingo
(b) the bearing of Yola from Jalingo.

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The town of Numan is 104 km from Jalingo on a bearing of North East. Yola is 56 km from Numan on a bearing of South . Sketch a diagram showing the points of the three towns. Calculate: (a) the distance of Yola from Jalingo (b) the bearing of Yola from Jalingo.

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Latest Geometry Questions

The town of Numan is 104 km from Jalingo
on a bearing of North East. Yola is 56 km
from Numan on a bearing of South .
Sketch a diagram showing the points of the
three towns. Calculate:
(a) the distance of Yola from Jalingo
(b) the bearing of Yola from Jalingo.