Question

what is the value of s?

Explanation:

Step1: Identify angle relationship

Since \( GH \perp HI \) and \( GF \perp FI \), and \( HI = FI = 74 \), triangles \( GHI \) and \( GFI \) are congruent (HL theorem). So \( \angle HGI=\angle FGI \), and \( \angle HGF = \angle HGI+\angle FGI=(s + 26^\circ)+2s \). Also, \( \angle HGF \) is a right angle? Wait, no, actually, since \( GH \) and \( GF \) are both perpendicular to \( HI \) and \( FI \) respectively, and \( HI = FI \), \( GI \) is the angle bisector? Wait, more accurately, the two right triangles \( GHI \) and \( GFI \) have hypotenuse \( GI \) common and leg \( HI = FI \), so they are congruent. Thus, \( \angle HGI=\angle FGI \), and the sum of \( \angle HGI \) and \( \angle FGI \) plus... Wait, actually, looking at the angles at \( G \), \( (s + 26^\circ)+2s = 90^\circ \)? Wait, no, maybe the two angles \( s + 26^\circ \) and \( 2s \) add up to \( 90^\circ \)? Wait, let's re - examine.

Wait, the figure has two right angles at \( H \) and \( F \), and \( HI = FI \), \( GI \) is common. So \( \triangle GHI\cong\triangle GFI \) (HL: right angle, hypotenuse \( GI \), leg \( HI = FI \)). Therefore, \( \angle HGI=\angle FGI \)? No, wait, \( \angle HGI = s + 26^\circ \) and \( \angle FGI=2s \), and since \( \triangle GHI\cong\triangle GFI \), the angles \( \angle HGI \) and \( \angle FGI \) should be such that their sum is \( 90^\circ \)? Wait, no, maybe the two angles at \( G \) ( \( s + 26^\circ \) and \( 2s \)) are complementary? Wait, let's set up the equation.

Since the two right triangles are congruent, the angles \( \angle HGI \) and \( \angle FGI \) should satisfy \( (s + 26^\circ)+2s=90^\circ \)? Wait, no, maybe the sum of these two angles is \( 90^\circ \)? Wait, let's think again. If we consider that \( GH \) and \( GF \) are both perpendicular to lines \( HI \) and \( FI \) which are equal in length, then \( GI \) bisects the angle between \( GH \) and \( GF \)? No, maybe the two angles \( s + 26 \) and \( 2s \) add up to \( 90^\circ \). Let's set up the equation:

\( (s + 26)+2s=90 \)

Step2: Solve the equation

Combine like terms: \( 3s+26 = 90 \)

Subtract 26 from both sides: \( 3s=90 - 26=64 \)? Wait, no, 90 - 26 is 64? Wait, 90 - 26 = 64? No, 90-26 = 64? Wait, 26+64 = 90, yes. Wait, but then \( s=\frac{64}{3}\)? That can't be right. Wait, maybe I made a mistake in the angle relationship.

Wait, maybe the two angles \( s + 26^\circ \) and \( 2s \) are such that they are equal? No, because \( \triangle GHI\cong\triangle GFI \), so \( \angle HGI=\angle FGI \)? Wait, no, \( \angle HGI = s + 26 \), \( \angle FGI = 2s \), so if they are equal, \( s + 26=2s \), then \( s = 26 \). But that doesn't seem to fit. Wait, maybe the sum of the two angles is \( 90^\circ \)? Wait, let's check the figure again. The right angles are at \( H \) and \( F \), so \( \angle GHI = 90^\circ \) and \( \angle GFI=90^\circ \), \( HI = FI = 74 \), \( GI \) is common. So by HL, \( \triangle GHI\cong\triangle GFI \), so \( \angle HGI=\angle FGI \), so \( s + 26=2s \), then \( s = 26 \). But then let's check the sum: \( 26 + 26+26\times2=26 + 26+52 = 104 eq90 \). So that's wrong.

Wait, maybe the two angles \( s + 26 \) and \( 2s \) are complementary, i.e., their sum is \( 90^\circ \). So:

\( s + 26+2s=90 \)

\( 3s=90 - 26 \)

\( 3s = 64 \)

\( s=\frac{64}{3}\approx21.33 \). No, that doesn't seem right. Wait, maybe the angle \( s + 26 \) and \( 2s \) are such that \( s + 26+2s = 90 \), but maybe I misread the figure. Wait, maybe the right angle is at \( G \)? No, the right angles are at \( H \) and \( F \). Wait, another approach: since \( HI = FI \), \( GH\perp HI \), \( GF\perp FI \), so \( GI \) is the angle bisector of \( \angle HGF \), and \( \angle HGF = 90^\circ \)? Wait, no, \( \angle HGF \) is not necessarily \( 90^\circ \). Wait, maybe the two triangles are congruent, so \( \angle HGI=\angle FGI \), so \( s + 26=2s \), so \( s = 26 \). But then let's check the sum of angles in triangle \( GHI \). In \( \triangle GHI \), right - angled at \( H \), angles are \( 90^\circ \), \( s + 26 \), and the third angle. If \( s = 26 \), then \( s + 26=52 \), so the third angle is \( 180 - 90 - 52 = 38 \). In \( \triangle GFI \), right - angled at \( F \), angle at \( G \) is \( 2s = 52 \), so the third angle is \( 180 - 90 - 52 = 38 \), which is consistent. Wait, but then why is the sum of \( s + 26 \) and \( 2s \) not \( 90 \)? Because \( \angle HGF=\angle HGI+\angle FGI=(s + 26)+2s \), and \( \angle HGF \) is not necessarily \( 90^\circ \). Wait, maybe I was wrong about the angle sum. Let's start over.

Since \( HI = FI \), \( \angle GHI=\angle GFI = 90^\circ \), and \( GI = GI \), by HL congruence, \( \triangle GHI\cong\triangle GFI \). Therefore, \( \angle HGI=\angle FGI \), so \( s + 26=2s \).

Step3: Solve \( s + 26=2s \)

Subtract \( s \) from both sides: \( 26=2s - s \)

So \( s = 26 \). Wait, but let's check again. If \( s = 26 \), then \( \angle HGI=26 + 26 = 52^\circ \), \( \angle FGI=2\times26 = 52^\circ \), so they are equal, which makes sense for congruent triangles. So the equation is \( s + 26=2s \), solving gives \( s = 26 \).

Wait, maybe my initial thought about the sum being \( 90^\circ \) was wrong. The key is that the two triangles are congruent, so their corresponding angles (the angles at \( G \)) are equal. So \( \angle HGI=\angle FGI \), hence \( s + 26=2s \).

Answer:

\( 26 \)