A cable is tied from point G to point K on a sign FH. Point K is halfway up the sign FH. The distance GH is 1.4 m and the height FH is 3 m. Mairéad claims that the angle $∠KGH$ is half the size of the angle $∠FGH$. We need to determine if Mairéad is correct.

1. Problem Description

A cable is tied from point G to point K on a sign FH. Point K is halfway up the sign FH. The distance GH is 1.4 m and the height FH is 3 m. Mairéad claims that the angle $∠KGH$ is half the size of the angle $∠FGH$. We need to determine if Mairéad is correct.

2. Solution Steps

First, we calculate the angle $∠FGH$ using the tangent function:

$tan(∠FGH) = \frac{FH}{GH} = \frac{3}{1.4}$

$∠FGH = arctan(\frac{3}{1.4})$

$∠FGH ≈ arctan(2.1429)$

$∠FGH ≈ 64.97°$

Next, we calculate the angle $∠KGH$. Since K is halfway up FH, the length KH is $\frac{3}{2} = 1.5$ m.

$tan(∠KGH) = \frac{KH}{GH} = \frac{1.5}{1.4}$

$∠KGH = arctan(\frac{1.5}{1.4})$

$∠KGH ≈ arctan(1.0714)$

$∠KGH ≈ 47.07°$

Now we check if $∠KGH$ is half of $∠FGH$:

$\frac{∠FGH}{2} = \frac{64.97}{2} ≈ 32.485°$

Since $∠KGH ≈ 47.07°$, it is not half of $∠FGH$.

3. Final Answer

No, Mairéad is not correct.