Math Problem Statement

plz solve these manually woth answers

For each of the following triangles find the length of the unknown lettered side (correct to two decimal places) or the size of the lettered angle (correct to one decimal place).

Solution

Let's work through each triangle step by step.

Problem (a):

We are given a triangle with:

  • ABC=52\angle ABC = 52^\circ,
  • AC=6cmAC = 6 \, \text{cm},
  • AE=12cmAE = 12 \, \text{cm},
  • A right angle at CC, and
  • The unknown side y=BCy = BC.

Using the sine rule in the right-angled triangle ABC\triangle ABC: sin(52)=ACBC=6y\sin(52^\circ) = \frac{AC}{BC} = \frac{6}{y} y=6sin(52)y = \frac{6}{\sin(52^\circ)} Let's calculate yy.

Problem (b):

We are given a right-angled triangle with:

  • AB=3mAB = 3 \, \text{m},
  • AD=15mAD = 15 \, \text{m},
  • CD=9mCD = 9 \, \text{m},
  • ABE=y\angle ABE = y^\circ (unknown angle), and
  • Right angles at BB and CC.

We can use the tangent rule to find the unknown angle yy: tan(y)=ABBD=39\tan(y) = \frac{AB}{BD} = \frac{3}{9} y=tan1(39)y = \tan^{-1}\left(\frac{3}{9}\right) Let's calculate yy.

Problem (c):

We have a circle and the triangle PQRPQR, where:

  • PQ=3cmPQ = 3 \, \text{cm},
  • PR=5cmPR = 5 \, \text{cm},
  • QR=6cmQR = 6 \, \text{cm}, and
  • PRQ=x\angle PRQ = x^\circ (unknown).

Using the cosine rule: cos(x)=PR2+PQ2QR22PRPQ\cos(x) = \frac{PR^2 + PQ^2 - QR^2}{2 \cdot PR \cdot PQ} Let's calculate xx.

Problem (d):

We have:

  • BE=8.6cmBE = 8.6 \, \text{cm},
  • AE=5.6cmAE = 5.6 \, \text{cm},
  • D=75\angle D = 75^\circ,
  • E=45\angle E = 45^\circ, and
  • x=BDx = BD is the unknown side.

We can use the sine rule to find xx: BEsin(45)=xsin(75)\frac{BE}{\sin(45^\circ)} = \frac{x}{\sin(75^\circ)} x=BEsin(75)sin(45)x = \frac{BE \cdot \sin(75^\circ)}{\sin(45^\circ)} Let's calculate xx.

Now, I will calculate all the unknowns.Here are the calculated results for each triangle:

(a)

The length of side BCBC (denoted yy) is approximately 7.61 cm.

(b)

The unknown angle yy^\circ is approximately 18.4°.

(c)

The angle xx^\circ in triangle PQRPQR is approximately 93.8°.

(d)

The length of side BDBD (denoted xx) is approximately 11.75 cm.

Would you like further details on these solutions, or have any specific questions?

Here are 5 related questions for further practice:

  1. How would you solve triangle (a) using the cosine rule instead?
  2. Can you use the sine rule to find the missing angles in triangle (b)?
  3. How would the solution change in problem (c) if one side length were altered?
  4. What is the general formula for solving triangles using the tangent rule?
  5. How can the sine rule be applied to solve non-right-angled triangles?

Tip: The sine and cosine rules are most useful when you have non-right-angled triangles or when you know two angles or two sides.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine Rule
Cosine Rule
Tangent Rule
Geometry of Circles

Formulas

Sine Rule: sin(A)/a = sin(B)/b = sin(C)/c
Cosine Rule: c^2 = a^2 + b^2 - 2ab * cos(C)
Tangent Rule: tan(θ) = opposite/adjacent

Theorems

Pythagorean Theorem
Tangent of an Angle
Cosine Rule for Non-Right-Angled Triangles
Sine Rule for Angles and Sides

Suitable Grade Level

Grades 9-12