https://file.aiquickdraw.com/mathbot/file/1727315166617znv1ycgy.jpg

To solve for the perimeter and area of the isosceles triangle shown in the image, we use the given side lengths:

- The two equal sides (legs) are each 34 mm.
- The base is 19 mm.

### Part (a): Perimeter
The perimeter \(P\) of a triangle is the sum of its side lengths. For this isosceles triangle:
\[ P = 34 \, \text{mm} + 34 \, \text{mm} + 19 \, \text{mm} = 87 \, \text{mm} \]

### Part (b): Area
To find the area \(A\) of the triangle, we need the height \(h\), which can be determined using the Pythagorean theorem on one-half of the triangle (splitting it into two right triangles).

1. **Calculate the height \(h\)**:
   - The hypotenuse is one of the legs of the triangle, 34 mm.
   - Half of the base is \(\frac{19 \, \text{mm}}{2} = 9.5 \, \text{mm}\).

   Apply the Pythagorean theorem:
   \[ h = \sqrt{(34 \, \text{mm})^2 - (9.5 \, \text{mm})^2} \]
   \[ h = \sqrt{1156 - 90.25} \]
   \[ h = \sqrt{1065.75} \]
   \[ h \approx 32.65 \, \text{mm} \]

2. **Calculate the area \(A\)**:
   \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
   \[ A = \frac{1}{2} \times 19 \, \text{mm} \times 32.65 \, \text{mm} \]
   \[ A = 

Use appropriate formulas to find (a) the perimeter and (b) the area of the figure.

Discover how to calculate the perimeter and area of an isosceles triangle using side lengths of 34 mm, 34 mm, and 19 mm. This guide provides step-by-step instructions utilizing the Pythagorean theorem to determine the height necessary for the area calculation. Suitable for students in Grades 7-9.

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