Find BC Round to the Nearest Tenth: A Step‑by‑Step Guide for Geometry Students
When working with triangles, one of the most frequent tasks is to determine the length of a specific side and then present that length rounded to a convenient precision. Even so, the phrase find bc round to the nearest tenth appears often in homework assignments, standardized tests, and real‑world applications such as engineering drafts or architectural plans. Which means mastering this skill not only boosts your grades but also sharpens your spatial reasoning. Below you’ll find a thorough explanation of the concepts, the procedural steps, and plenty of practice examples to help you become confident in finding side BC and rounding the result correctly.
1. What Does “Find BC Round to the Nearest Tenth” Mean?
In a triangle labeled △ABC, the vertices are typically named A, B, and C. The side opposite vertex A is BC, the side opposite B is AC, and the side opposite C is AB. When a problem asks you to find bc round to the nearest tenth, it is requesting:
- Calculate the exact length of side BC using the appropriate geometric relationship (Pythagorean theorem, Law of Cosines, Law of Sines, etc.).
- After obtaining the exact or decimal value, round that number to one decimal place (the nearest tenth).
Rounding to the nearest tenth simplifies communication while retaining sufficient accuracy for most practical purposes The details matter here. Worth knowing..
2. Core Formulas You’ll Need
Depending on the information given, different formulas apply. Keep these handy:
| Situation | Formula | When to Use |
|---|---|---|
| Right triangle, legs known | (BC = \sqrt{AB^2 + AC^2}) (if BC is the hypotenuse) or (BC = \sqrt{ | AB^2 - AC^2 |
| Any triangle, two sides and included angle known | (BC^2 = AB^2 + AC^2 - 2·AB·AC·cos(∠A)) | You know sides AB, AC and the angle between them (∠A). |
| Any triangle, one side and two angles known | (\displaystyle \frac{BC}{\sin(∠A)} = \frac{AB}{\sin(∠C)} = \frac{AC}{\sin(∠B)}) | You know at least one side length and its opposite angle, plus another angle. |
| Any triangle, all three sides known (to verify) | Use the Law of Cosines to solve for an angle, then apply the sine law if needed. | You need to check consistency or find an angle first. |
Remember: after you compute BC, apply the rounding rule: look at the hundredths digit; if it is 5 or greater, increase the tenths digit by one; otherwise, leave the tenths digit unchanged and drop the rest.
3. Step‑by‑Step Procedure to Find BC and Round to the Nearest Tenth
Follow this checklist whenever you encounter a problem that asks you to find bc round to the nearest tenth:
-
Identify the known quantities
- List all given side lengths and angle measures.
- Determine which side is BC (the side opposite vertex A).
-
Choose the appropriate formula
- If a right angle is present → Pythagorean theorem.
- If you have two sides and the angle between them → Law of Cosines.
- If you have one side and its opposite angle plus another angle → Law of Sines.
-
Set up the equation
- Substitute the known values into the chosen formula.
- Keep the equation in a form that isolates (BC^2) or (BC).
-
Solve for BC
- Perform the arithmetic carefully.
- Take square roots when needed, and remember that a length is always positive.
-
Round the result
- Look at the digit in the hundredths place.
- Apply the rounding rule to obtain the value to the nearest tenth.
-
State the answer clearly
- Include the unit (if any) and the phrase “rounded to the nearest tenth” to show you followed instructions.
4. Worked Examples
Example 1: Right Triangle – Using the Pythagorean Theorem
Problem: In right triangle △ABC, ∠B = 90°, AB = 6 cm, AC = 10 cm. Find BC rounded to the nearest tenth The details matter here..
Solution:
- Since ∠B is the right angle, the hypotenuse is AC. The legs are AB and BC.
- Apply the Pythagorean theorem: (AC^2 = AB^2 + BC^2).
- Rearranged: (BC^2 = AC^2 - AB^2 = 10^2 - 6^2 = 100 - 36 = 64).
- (BC = \sqrt{64} = 8) cm.
- The hundredths digit is 0 (since 8.00), so rounding to the nearest tenth leaves 8.0 cm.
Answer: BC = 8.0 cm (rounded to the nearest tenth).
Example 2: Oblique Triangle – Using the Law of Cosines
Problem: In triangle △ABC, AB = 7 in, AC = 9 in, and ∠A = 45°. Find BC rounded to the nearest tenth.
Solution:
- We have two sides and the included angle → Law of Cosines.
- Formula: (BC^2 = AB^2 + AC^2 - 2·AB·AC·cos(∠A)).
- Substitute: (BC^2 = 7^2 + 9^2 - 2·7·9·cos45°).
- Compute: (7^2 = 49), (9^2 = 81); (2·7·9 = 126); (\cos45° = \frac{\sqrt{2}}{2} ≈ 0.7071).
- (BC^2 = 49 + 81 - 126·0.7071 = 130 - 89.09 ≈ 40.91).
- (BC = \sqrt{40.91} ≈ 6.395) in.
- Look at the hundredths digit: 9 (≥5), so increase the tenths digit from 3 to 4.
- Rounded value: 6.4 in.
Answer: BC ≈ 6.4 in (rounded to the nearest tenth).
Example 3: Using the Law of Sines
Problem: In triangle △ABC, ∠A = 30°, ∠B = 70°, and side AB = 12 m. Find BC rounded to the nearest tenth
