Question

given right triangle jkm, which correctly describes the locations of the sides in relation to ∠j?
○ a is the hypotenuse, b is adjacent, c is opposite
○ a is the hypotenuse, b is opposite, c is adjacent
○ a is adjacent, b is opposite, c is the hypotenuse
○ a is opposite, b is the hypotenuse, c is adjacent

Explanation:

Step1: Recall triangle side definitions

In a right triangle, hypotenuse is opposite right angle. For an acute angle (∠J), opposite side is across, adjacent is next to, hypotenuse is longest (side \(a\) here, as it's opposite right angle at \(M\)).

Step2: Analyze sides relative to ∠J

- Hypotenuse: \(a\) (opposite right angle at \(M\)).
- Opposite to ∠J: side \(c\) (across from ∠J).
- Adjacent to ∠J: side \(b\) (next to ∠J, shares ∠J's vertex). Wait, no—wait, ∠J is at vertex J. So:
- Hypotenuse: \(a\) (JK, opposite right angle M).
- Opposite ∠J: side \(c\) (KM, across from J).
- Adjacent ∠J: side \(b\) (JM, next to J). Wait, no, the options: let's re - evaluate. Wait, the options: second option says "a is the hypotenuse, b is opposite, c is adjacent". Wait, ∠J: side opposite ∠J is KM (length \(c\))? No, wait, vertex J: the sides:
- Hypotenuse: \(a\) (JK, since right angle at M, so JK is hypotenuse).
- Opposite ∠J: side \(c\) (KM) or \(b\) (JM)? Wait, ∠J is at J, so the sides forming ∠J are JM (length \(b\)) and JK (length \(a\)). The side opposite ∠J is KM (length \(c\))? No, wait, no—wait, in triangle JKM, right - angled at M. So:
- Hypotenuse: JK (\(a\)), as it's opposite the right angle (∠M).
- For ∠J:
- Adjacent side: JM (\(b\)), because it is one of the sides that form ∠J (along with JK).
- Opposite side: KM (\(c\)), because it is across from ∠J (not forming ∠J). Wait, but the second option is "a is the hypotenuse, b is opposite, c is adjacent". Wait, maybe I mixed up. Wait, ∠J: the side opposite ∠J is KM (length \(c\))? No, wait, vertex J: the sides:
- The two legs: JM (\(b\)) and KM (\(c\)). Hypotenuse: JK (\(a\)).
- For angle J:
- Adjacent side: JM (\(b\))? No, adjacent side is the leg that is part of angle J. Angle J is between JM and JK. So JM is adjacent, KM is opposite. Hypotenuse is JK (\(a\)). Wait, the second option: "a is the hypotenuse, b is opposite, c is adjacent". Wait, if b is JM, opposite angle J would be KM (c)? No, this is confusing. Wait, let's re - express:

In right triangle JKM, right - angled at M. So:

- Hypotenuse: JK (\(a\)) (opposite right angle M).

- For angle J:

- Opposite side: KM (\(c\))? No, wait, angle J is at J. The side opposite angle J is the side that does not have vertex J. So side KM (length \(c\)) has vertices K and M, so it's opposite angle J.

- Adjacent side: JM (length \(b\)) has vertex J, so it's adjacent to angle J.

But the second option says "a is the hypotenuse, b is opposite, c is adjacent". Wait, maybe I got the labels wrong. Wait, the triangle: K---c---M, M---b---J, J---a---K. So angle at J: between J - M and J - K. So:

- Hypotenuse: J - K (\(a\)) (opposite right angle at M).

- Opposite angle J: K - M (\(c\))? No, K - M is opposite angle J? Wait, angle at J: the side opposite is K - M (since angle at J, the side not connected to J is K - M). The side adjacent is J - M (connected to J and part of the angle).

But the second option is "a is the hypotenuse, b is opposite, c is adjacent". So if b is J - M, opposite angle J would be K - M (c)? No, that's not. Wait, maybe the labels are: side a is JK, side b is JM, side c is KM.

So for angle J:

- Hypotenuse: a (JK, opposite right angle M).

- Opposite angle J: side c (KM, because it's across from J).

- Adjacent angle J: side b (JM, because it's next to J, forming angle J with JK).

Wait, but the second option says "b is opposite, c is adjacent". So maybe I have the opposite and adjacent reversed. Wait, angle J: the adjacent side is the one that is a leg and is adjacent to angle J (so JM, length b), and the opposite side is the leg that is opposite (KM, length c). But the second option says "a is the hypotenuse, b is opposite, c is adjacent". So if we consider angle J, the side opposite angle J is JM (b)? No, that can't be. Wait, maybe the triangle is labeled differently. Wait, the right angle is at M, so vertices are J, K, M with right angle at M. So sides:

- JM: length b (vertical leg).

- KM: length c (horizontal leg).

- JK: length a (hypotenuse).

Angle at J: between JM (b) and JK (a). So:

- Hypotenuse: a (JK).

- Opposite angle J: KM (c) (because it's not connected to J).

- Adjacent angle J: JM (b) (connected to J and part of the angle).

But the second option is "a is the hypotenuse, b is opposite, c is adjacent". So this is a contradiction unless my understanding is wrong. Wait, maybe "opposite" and "adjacent" are with respect to angle J, but I mixed up the sides. Wait, let's check the options again:

Option 2: "a is the hypotenuse, b is opposite, c is adjacent".

If angle J:

- Hypotenuse: a (correct, as it's opposite right angle M).

- Opposite angle J: side b (JM) – no, that's adjacent. Wait, no, maybe the angle is at K? No, the question is about angle J.

Wait, maybe I made a mistake. Let's recall: in a right triangle, for an acute angle:

- Hypotenuse: the side opposite the right angle (longest side).

- Opposite side: the side across from the acute angle (not forming the angle).

- Adjacent side: the side that is part of the acute angle (along with the hypotenuse).

So for angle J:

- Hypotenuse: a (JK, opposite right angle M).

- Opposite side: KM (c) (across from J).

- Adjacent side: JM (b) (part of angle J, along with JK).

But the second option says "b is opposite, c is adjacent". So this is wrong. Wait, the first option: "a is the hypotenuse, b is adjacent, c is opposite" – that matches our analysis. Wait, no, first option: "a is the hypotenuse, b is adjacent, c is opposite". Let's re - check:

- Hypotenuse: a (correct).

- Adjacent to J: b (JM, correct, as it's part of angle J).

- Opposite to J: c (KM, correct, as it's across from J).

Wait, no, the first option is "a is the hypotenuse, b is adjacent, c is opposite" – that's correct. But wait, the second option is "a is the hypotenuse, b is opposite, c is adjacent". I think I messed up earlier. Let's start over:

Right triangle JKM, right - angled at M. So:

- Vertices: J, K, M. Right angle at M.

- Sides:

- JK: length a (hypotenuse, opposite right angle M).

- JM: length b (leg, connects J and M).

- KM: length c (leg, connects K and M).

Angle at J: between sides JM (b) and JK (a).

- Hypotenuse: a (JK) – correct.

- Adjacent to angle J: JM (b) – because it is one of the sides forming angle J (along with JK).

- Opposite to angle J: KM (c) – because it is the side that does not form angle J (it is across from angle J).

So the first option: "a is the hypotenuse, b is adjacent, c is opposite" – this is correct? Wait, no, the first option says "c is opposite". Wait, KM is length c, which is opposite angle J. JM is length b, adjacent to angle J. JK is length a, hypotenuse. So first option: "a is the hypotenuse, b is adjacent, c is opposite" – that's correct. But wait, the second option: "a is the hypotenuse, b is opposite, c is adjacent" – let's see: if b is opposite, that would mean JM is opposite angle J, but JM is adjacent. If c is adjacent, that would mean KM is adjacent, but KM is opposite. So the first option is correct? Wait, no, maybe I have the labels of the sides wrong. Wait, the diagram: K to M is c, M to J is b, J to K is a. So angle at J: the sides are JM (b) and JK (a). The side opposite angle J is KM (c). The side adjacent is JM (b). So the first option: "a is the hypotenuse, b is adjacent, c is opposite" – that's correct. But wait, the second option is "a is the hypotenuse, b is opposite, c is adjacent". I think I made a mistake in the initial analysis. Wait, let's check the options again:

Option 1: a is hypotenuse, b is adjacent, c is opposite.

Option 2: a is hypotenuse, b is opposite, c is adjacent.

Option 3: a is adjacent, b is opposite, c is hypotenuse. (Wrong, a is hypotenuse)

Option 4: a is opposite, b is hypotenuse, c is adjacent. (Wrong, a is hypotenuse)

So between option 1 and 2. Let's re - define opposite and adjacent for angle J:

- Opposite side: the side that does not contain vertex J. So side KM (c) does not contain J, so it's opposite angle J.

- Adjacent side: the side that contains vertex J and is not the hypotenuse. So side JM (b) contains J and is a leg, so it's adjacent to angle J.

- Hypotenuse: side JK (a) contains J and is the hypotenuse.

So option 1: "a is the hypotenuse, b is adjacent, c is opposite" – this is correct. Wait, but the original problem's second option – maybe I misread the labels. Wait, maybe the side labeled b is KM and c is JM? No, the diagram shows K---c---M, M---b---J, J---a---K. So M to J is b, K to M is c. So angle J: adjacent is M to J (b), opposite is K to M (c), hypotenuse is J to K (a). So option 1 is correct. But wait, the user's options: let's check again.

Wait, the options are:

1. a is the hypotenuse, b is adjacent, c is opposite

2. a is the hypotenuse, b is opposite, c is adjacent

3. a is adjacent, b is opposite, c is the hypotenuse

4. a is opposite, b is the hypotenuse, c is adjacent

So according to our analysis, option 1 is correct? Wait, no, wait: angle J. The side opposite angle J is the side that is not connected to J. So side KM (c) is not connected to J, so it's opposite. Side JM (b) is connected to J, so it's adjacent. Hypotenuse is JK (a). So option 1: "a is hypotenuse, b is adjacent, c is opposite" – correct. But wait, maybe the problem has a different labeling. Wait, maybe the right angle is at K? No, the diagram shows right angle at M (square at M). So right angle at M. So our analysis holds. So the correct option is the first one? Wait, no, wait, I think I made a mistake. Let's think about angle J:

In triangle JKM, right - angled at M. So angle at J: the sides:

- Hypotenuse: JK (a) (opposite right angle M).

- Opposite angle J: KM (c) (because it's across from J).

- Adjacent angle J: JM (b) (because it's next to J, forming angle J with JK).

So the first option: "a is the hypotenuse, b is adjacent, c is opposite" – this is correct. But wait, the second option: "a is the hypotenuse, b is opposite, c is adjacent" – if we consider angle J, b is JM, which is adjacent, not opposite. c is KM, which is opposite, not adjacent. So the first option is correct.

Wait, but maybe I have the opposite and adjacent reversed. Let's recall the definition:

- Adjacent side: the side that is part of the angle (other than the hypotenuse).

- Opposite side: the side that is not part of the angle.

So for angle J, the angle is formed by sides JM (b) and JK (a). So the side that is part of the angle (other than hypotenuse) is JM (b) – adjacent. The side not part of the angle is KM (c) – opposite. Hypotenuse is JK (a). So option 1 is correct.

Answer:

A. a is the hypotenuse, b is adjacent, c is opposite