Question: (b) For each of the following triangles, determine whether the triangle is acute-angled or obtuse-angled. If the triangle is obtuse-angled, state which angle is the obtuse angle.
(1) In $\Delta ABC$, $AB = 24$, $BC = 14$ and $AC = 27$.
(2) In $\Delta DEF$, $d = 8$ cm, $e = 18$ cm and $f = 13$ cm.
(3) In $\Delta PQR$, $PQ = 17$ cm, $QR = 13$ cm and $PR = 10$ cm.
(4) In $\Delta LMN$, $LM = 5$, $MN = 12$ and $LN = 11$.
1. **Problem Statement:** Determine if each triangle is acute-angled or obtuse-angled using the side lengths given.
2. **Formula and Rule:** Use the Law of Cosines or compare squares of sides to classify the triangle.
For sides $a$, $b$, and $c$ with $c$ the longest side:
- If $a^2 + b^2 > c^2$, the triangle is acute-angled.
- If $a^2 + b^2 = c^2$, the triangle is right-angled.
- If $a^2 + b^2 < c^2$, the triangle is obtuse-angled, and the angle opposite side $c$ is obtuse.
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### (1) Triangle $\Delta ABC$ with sides $AB=24$, $BC=14$, $AC=27$
- Longest side: $AC=27$
- Calculate squares:
$$AB^2 + BC^2 = 24^2 + 14^2 = 576 + 196 = 772$$
$$AC^2 = 27^2 = 729$$
- Compare:
$$772 > 729$$
- Since $AB^2 + BC^2 > AC^2$, the triangle is **acute-angled**.
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### (2) Triangle $\Delta DEF$ with sides $d=8$, $e=18$, $f=13$
- Longest side: $e=18$
- Calculate squares:
$$d^2 + f^2 = 8^2 + 13^2 = 64 + 169 = 233$$
$$e^2 = 18^2 = 324$$
- Compare:
$$233 < 324$$
- Since $d^2 + f^2 < e^2$, the triangle is **obtuse-angled**.
- The obtuse angle is opposite the longest side $e$, so angle $E$ is obtuse.
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### (3) Triangle $\Delta PQR$ with sides $PQ=17$, $QR=13$, $PR=10$
- Longest side: $PQ=17$
- Calculate squares:
$$QR^2 + PR^2 = 13^2 + 10^2 = 169 + 100 = 269$$
$$PQ^2 = 17^2 = 289$$
- Compare:
$$269 < 289$$
- Since $QR^2 + PR^2 < PQ^2$, the triangle is **obtuse-angled**.
- The obtuse angle is opposite the longest side $PQ$, so angle $P$ is obtuse.
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### (4) Triangle $\Delta LMN$ with sides $LM=5$, $MN=12$, $LN=11$
- Longest side: $MN=12$
- Calculate squares:
$$LM^2 + LN^2 = 5^2 + 11^2 = 25 + 121 = 146$$
$$MN^2 = 12^2 = 144$$
- Compare:
$$146 > 144$$
- Since $LM^2 + LN^2 > MN^2$, the triangle is **acute-angled**.
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**Final answers:**
1. $\Delta ABC$ is acute-angled.
2. $\Delta DEF$ is obtuse-angled; obtuse angle is $\angle E$.
3. $\Delta PQR$ is obtuse-angled; obtuse angle is $\angle P$.
4. $\Delta LMN$ is acute-angled.